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Theorem elon 4591
Description: An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
Hypothesis
Ref Expression
elon.1  |-  A  e. 
_V
Assertion
Ref Expression
elon  |-  ( A  e.  On  <->  Ord  A )

Proof of Theorem elon
StepHypRef Expression
1 elon.1 . 2  |-  A  e. 
_V
2 elong 4590 . 2  |-  ( A  e.  _V  ->  ( A  e.  On  <->  Ord  A ) )
31, 2ax-mp 8 1  |-  ( A  e.  On  <->  Ord  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    e. wcel 1726   _Vcvv 2957   Ord word 4581   Oncon0 4582
This theorem is referenced by:  tron  4605  0elon  4635  smogt  6630  rdglim2  6691  omeulem1  6826  isfinite2  7366  r0weon  7895  cflim3  8143  inar1  8651  ellimits  25756  dford3lem2  27099
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ral 2711  df-rex 2712  df-v 2959  df-in 3328  df-ss 3335  df-uni 4017  df-tr 4304  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586
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